One of the largest cities in the world, with 12 million people, there are a lot of things to see and do in Seoul. If you like history, there are three royal palaces - Toksugung, Changdokkung, and Kyobokkung. The National War Memorial Museum provides an interesting view of weapons and armor on the Korean peninsula from the bronze age to the Korean War - there's even a B-52 and MiG-19 on display.
Namdaemun is a marketplace filled with everything from food, clothing, housewares and electronics. Tongademun marketplace sells mostly clothing and is the place to go for running shoes. Insadong is great for antiques, pottery and lacquerware. A lot of nightlife, clubs, etc. are located in Kangnam. If you like panoramic views, Namsan Hill and Seoul Tower will give you excellent views of the City.
The subway system is extensive, clean and reasonably priced with signs in English as well as Korean Hangul. Of course, there is always the DMZ and organized tour of Panmunjon - you cannot go there independently.